上同调方程解的Sobolev正则性

Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps

We prove that the solutions of the cohomological equation for Roth type interval exchange maps are Hölder continuous provided that the datum is of class $C^r$ with $r>1$ and belongs to a finite-codimension linear subspace.

Pollicott–Ruelle预解式与Sobolev正则性

Pollicott-Ruelle resolvent and Sobolev regularity

本文研究Anosov流的Pollicott–Ruelle预解式，在一般向量丛提升情形下确定其在各向异性Sobolev空间中亚纯延拓的正则性阈值，并联系关联衰减所需正则性。

The twisted cohomological equation over the geodesic flow

We study the twisted cohomoligical equation over the geodesic flow on $SL(2,\mathbb{R})/Γ$. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates for the solution, i.e, there is finite loss of regularity (with respect to Sobolev norms) between the twisted coboundary and the solution. We also give a tame splittings for non-homogeneous cohomological equations. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for partially hyperbolic actions in …

The Cohomological Equation and Invariant Distributions for Horocycle Maps

We study the invariant distributions for the horocycle map on $Γ\backslash SL(2, \mathbb{R})$ and prove Sobolev estimates for the cohomological equation of the horocycle map. As an application, we obtain an estimate for the rate of equidistribution for horocycle maps on compact manifolds.

多重混合与多重分数上同调方程：半单情形

Multiple mixing and multiple fractional cohomological equation: semisimple setting

本文提出了一种新的有效方法，用于半单情形下的高阶混合，证明了在强谱间隙假设下，部分超bolic代数作用的所有阶次的有效指数混合，其衰减率明确依赖于Lyapunov和谱间隙数据，所需Sobolev阶数也明确。在二级估计中，仅需要部分Sobolev/Hölder正则性沿弱稳定和不稳定子群方向，无需横截导数。对于允许比温和衰减更好的表示，所得二级估计达到最优矩阵系数指数。

求解局部哈密顿流的上同调方程，第一部分——局部障碍

Solving the cohomological equation for locally hamiltonian flows, part I -- local obstructions

本文研究紧致曲面上局部哈密顿流的余同调方程，通过定义新的不变分布族（局部障碍）并分析其影响，给出了解的正则性条件，为完整解决该问题提供关键要素。